On a toric manifold \(H^2(M, =H^{1,1}(M, \) , so we are in the case \(\alpha =\overline{\alpha }\)  with \(F^{[m]}=\overline{\omega }_{\varphi }^{[m]}\) . By (REF ) and using (REF ), we compute
\(\begin{split}{\bf M}(F) &= {\bf M}(\overline{\omega }_{\varphi }) +(2\pi )^m \int _{{\mathrm {P}}} \log \left(\frac{\det \mathrm {Hess}(u)}{\det (\mathrm {Hess}(u) + \sqrt{-1}B)}\right) {\mathrmsl d}x.\end{split}\) 
The claim follows by the above, taking in mind that \(\overline{\omega }_{\varphi } = {\mathrmsl d}{\mathrmsl d}^c_{J_A} \phi (\tilde{y})\) , where \(\phi \)  is the Legendre transform of \(u\in {\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\)  (see ()) and \(\tilde{y}\)  are the pluriharmonic coordinates of \(J_A\)  (see (REF )), and the fact that (see [1]})
\({\bf M}(\overline{\omega }_{\varphi }) = (2\pi )^m\left({\bf F}(u) -\int _{{\mathrm {P}}} \log \det (\mathrm {Hess}(u)) {\mathrmsl d}x + \int _{{\mathrm {P}}} \log \det (\mathrm {Hess}(u_0)){\mathrmsl d}x \right) .\) 
